Can an Abelian Group Be Isomorphic to a Non-Abelian Group in Physics?

[Konte]

Hi everybody,

I have a question: is an abelian group can be isomorphic to a non-abelian group?

Thank you everybody.

 

[DrDu]

In a non-abelian group, there are at least two elements A and B so that A*B=C but B*A=D with different D and C. this is not possible in an abelian group, so the two groups can also not be isomorphic.

 

[Konte]

In a non-abelian group, there are at least two elements A and B so that A*B=C but B*A=D with different D and C. this is not possible in an abelian group, so the two groups can also not be isomorphic.

I thought so. But I asked this question because I have read this article of G.Bone and Co. http://dx.doi.org/10.1080/00268979100100021(in appendix page 71-72), which states that the Molecular symmetry group (MS) of a rigid molecule is isomorphic to its point group. Knowing that the MS group is composed of nuclear permutation-inversion operation, I verified by myself this group is in general non-abelian, and on the other side, point group are in general abelian. So, did they say something wrong?

Thank you.

Konte

 

[DrDu]

Most point groups aren't abelian. Take a cubic group, like Oh as an example: Rotate first 90 deg, around x and then 90 deg. around y. Compare to first rotating 90 deg around y and then 90 deg around x.

 

[Konte]

You are right, I missed it.
Thanks a lot.

I have another question, how to construct the character table of irreducible representation knowing the multiplication table of the group?

 

[DrDu]

This can be arbitrarily nontrivial. Best have a look on a decent book on group theory. As you are mainly interested in the symmetric group, Sternberg, "Group theory and physics" might be what you are looking for.

 

[Konte]

Ok. Thanks!